Graduate Student Research
Permanent URI for this community
This community holds theses and dissertations submitted to the Faculty of Graduate Studies. For more information please consult the Faculty of Graduate Studies website http://www.uvic.ca/graduatestudies/resourcesfor/students/thesis/index.php
Browse
Browsing Graduate Student Research by Supervisor "Agueh, Martial"
Now showing 1 - 5 of 5
Results Per Page
Sort Options
Item Analysis of a mollified kinetic equation for granular media(2016-08-15) Thompson, William; Illner, Reinhard; Agueh, MartialWe study a nonlinear kinetic model describing the interactions of particles in a granular medium, i.e. inelastic systems where kinetic energy is not conserved due to internal friction. Examples of particles that fall into this category are sand, ground coffee and many others. Originally studied by Benedetto, Caglioti and Pulvirenti in the one-dimensional setting (RAIRO Model. Math. Anal. Numer, 31(5): 615-641, (1997)) the original model contained inconsistencies later accounted for and corrected by invoking a mollifier (Modelisation Mathematique et Analyse Numerique, M2AN, Vol. 33, No 2, pp. 439–441 (1999)). This thesis approximates the generalized model presented by Agueh (Arch. Rational Mech., Anal. 221, pp. 917-959 (2016)) with the added assumption of a spatial mollifier present in the kinetic equation. In dimension d ≥ 1 this model reads as ∂tf + v · ∇xf = divv(f([ηα∇W] ∗(x,v) f)) where f is a non-negative particle density function, W is a radially symmetric class C2 velocity interaction potential, and and ηα is a mollifier. A physical interpretation of this approximation is that the particles are spheres of radius α > 0 as opposed to the original assumption of being point-masses. Properties lost by this approximation and macroscopic quantities that remain conserved are discussed in greater detail and contrasted. The main result of this thesis is a proof of the weak global existence and uniqueness. An argument utilizing the tools of Optimal Transport allows simple construction of a weak solution to the kinetic model by transporting an initial measure under the characteristic flow curves. Concluding regularity arguments and restrictions on the velocity interaction potential ascertain that global classical solutions are obtained.Item An efficient numerical algorithm for the L2 optimal transport problem with applications to image processing(2010-12-13T22:10:24Z) Saumier Demers, Louis-Philippe; Agueh, Martial; Khouider, BoualemWe present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method relies on a numerical resolution of the corresponding Monge-Ampère equation. We use an existing Newton-like algorithm that we generalize to the case of a non uniform final density. The main idea consists of designing an iterative scheme where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we discretize and solve at each iteration, in two different ways: a second order finite difference scheme and a Fourier transform (FT) method. The FT method, made possible thanks to a preconditioning step based on the coefficient-averaged equation, results in an overall O(P LogP )-operations algorithm, where P is the number of discretization points. We prove that the generalized algorithm converges to the solution of the optimal transport problem, under suitable conditions on the initial and final densities. Numerical experiments demonstrating the robustness and efficiency of the method on several examples of image processing, including an application to multiple sclerosis disease detection, are shown. We also demonstrate by numerical tests that the method is competitive against some other methods available.Item The Mathematics of principal-agent problem with adverse selection(2011-08-19) Shadnam, Mojdeh; Agueh, Martial; Ye, Juan JuanThis thesis studies existence and characterization of optimal solutions to the principal-agent problem with adverse selection for both discrete and continuous problems. The existence results are derived by the abstract concepts of differentiability and convexity. Under the Spence Mirrlees condition, we show that the discrete problem reduces to a problem that always satisfies the linear independence constraint qualification, while the continuum of type problem becomes an optimal control problem. We then use the Ellipsoid algorithm to solve the problem in the discrete and convex case. For the problem without the Spence Mirrlees condition, we consider different classes of constraint qualifications. Then we introduce some easy-to-check conditions to verify these constraint qualifications. Finally we give economic interpretations for several numerical examples.Item Models for Particle Image Velocimetry: Optimal Transportation and Navier-Stokes Equations(2016-01-15) Saumier Demers, Louis-Philippe; Agueh, Martial; Khouider, BoualemWe introduce new methods based on the L2 Optimal Transport (OT) problem and the Navier-Stokes equations to approximate a fluid velocity field from images obtained with Particle Image Velocimetry (PIV) measurements. The main idea is to consider two successive images as the initial and final densities in the OT problem, and to use the associated OT flow as an estimate of the underlying physical flow. We build a simple but realistic model for PIV data, and use it to analyze the behavior of the transport map in this situation. We then design and implement a series of post-processing filters created to improve the quality of the numerical results, and we establish comparisons with traditional cross-correlation algorithms. These results indicate that the OT-PIV procedure performs well on low to medium seeding densities, and that it gives better results than typical cross-correlation algorithms in some cases. Finally, we use a variational method to project the OT velocity field on the space of solutions of the Navier-Stokes equations, and extend it to the rest of the fluid domain, outside the particle locations. This extension provides an effective filtering of the OT solution beyond the local post-processing filters, as demonstrated by several numerical experiments.Item Weak Solutions to a Fractional Fokker-Planck Equation via Splitting and Wasserstein Gradient Flow(2014-08-22) Bowles, Malcolm; Agueh, MartialIn this thesis, we study a linear fractional Fokker-Planck equation that models non-local (`fractional') diffusion in the presence of a potential field. The non-locality is due to the appearance of the `fractional Laplacian' in the corresponding PDE, in place of the classical Laplacian which distinguishes the case of regular (Gaussian) diffusion. Motivated by the observation that, in contrast to the classical Fokker-Planck equation (describing regular diffusion in the presence of a potential field), there is no natural gradient flow formulation for its fractional counterpart, we prove existence of weak solutions to this fractional Fokker-Planck equation by combining a splitting technique together with a Wasserstein gradient flow formulation. An explicit iterative construction is given, which we prove weakly converges to a weak solution of this PDE.